The full symmetric Toda system is a straightforward generalization of the usual (3-diagonal) system; it can be further generalized to the case of Cartan decomposition of an arbitrary real semisimple Lie group. In this case the integrability of the system is known, but the constructions of the involute families of integrals are usually quite complicated. In my talk I will describe a construction of commutative family of vector fields on the compact group, analogous to the family of first integrals in involution. This construction is based on the structure of representations of the original group. If time permits, I will also describe the relation of this construction with Sorin and Chernyakov’s and Reshetikhin and Schrader’s constructions, proving the noncommutative integrability of the system. If time permits, I shall also speak about another interesting aspect of the full symmetric Toda system: it turns out, that the phase portrait of this system is determined by an important discrete structure, the so called Bruhat order on the Weyl group of the corresponding Lie group. In my talk I will give the necessary definitions and sketch the proofs.
